Understanding Relationships Between Fractions, Decimals, Rat...

**1. **The Literacy and Numeracy Secretariat Professional Learning Series Understanding Relationships Between Fractions, Decimals, Ratios, Rates, and Percents

**2. **2 Session A ? Activating Mathematical Knowledge Aims of Numeracy Professional Learning
Learning Goals of the Module
Warm Up ? We Are Fractions!
Overview of Number Sense and Numeration, Grades 4 to 6
Scavenger Hunt ? Volume 1: The Big Ideas
Book Walk ? Volume 5: Fractions

**3. **3 Aims of Numeracy Professional Learning Promote the belief that all students have learned some mathematics through their lived experiences in the world and that the math classroom is one where students bring that thinking to their work.
Build teachers? expertise at setting classroom conditions where students can move from their informal math understandings to forming concepts, making sense of procedures and becoming comfortable with formal mathematical representations.
Assist educators working with teachers of students in the junior division to implement student-focused instructional methods to improve student achievement ? as referenced in Number Sense and Numeration, Grades 4-6.

**4. **4 Have teachers experience mathematical problem solving as a model of what effective math instruction entails by:
collectively solving problems relevant to students? lives that reflect the expectations in the Ontario mathematics curriculum;
viewing and discussing the thinking and strategies in the solutions;
sorting and classifying the responses to a problem to provide a visual image of the range of experience and understanding of the mathematics; and
analysing the visual continuum of thinking to determine starting points for instruction. Aims continued

**5. **5 Teaching Mathematics Through Problem Solving Sharing thinking
Listening to and considering ideas of others
Adapting thoughts
Understanding and analysing solutions
Comparing and contrasting different solutions
Discussing
Generalizing
Communicating

**6. **6 During these sessions, participants will:
develop an understanding of the conceptual models of fractions, decimals, ratios, rates, and percents;
explore conceptual and algorithmic models of fractions and decimals through problem solving;
analyse and discuss the role of student-generated strategies and standard algorithms in teaching the concepts and relationships of fractions, decimals, ratios, rates, and percents; and
identify, reflect on, and connect strategies that form a major component of an effective mathematics classroom. Learning Goals of the Module Share the session goals with the participants. Share the session goals with the participants.

**7. **7 Warm Up ? We Are Fractions! Introduce yourself to anyone at your table you do not know.
In your group, make a list of the following:
3 or 4 four things that might be true of nearly all of us
3 or 4 four things that might be true of nearly half of us
3 or 4 four things that might be true of nearly none of us This activity will help to ?break the ice? and introduce the topic of fractions.This activity will help to ?break the ice? and introduce the topic of fractions.

**8. **8 Scavenger Hunt ? Volume 1: The Big Ideas If participants have already received the guide, ask them ahead of time to bring it to the session. Give each participant a copy of BLM1.
If participants have already received the guide, ask them ahead of time to bring it to the session. Give each participant a copy of BLM1.

**9. **9 Session B ? Modelling and Representing Warm Up ? Anticipation Guide
What Does It Mean to Model and Represent Mathematical Thinking?
Save, Save, Save ? Problem #1
A Mini-Gallery Walk

**10. **10 Warm Up ? Anticipation Guide Talk to your table partners. Come up with a table answer for the following statements:

**11. **11 Save, Save, Save ? Problem #1 Patrik sees white shirts on sale. A sign in the window shows a 25% discount. Another sign shows different white shirts with off. A third sign shows discounted prices that are 0.45 less than the original price on white shirts.
Show Patrik which discount he should ask for in order to save the most money on a white shirt.
Show more than one way to solve this problem.

**12. **12 Solving the Problem Patrik sees white shirts on sale. A sign in the window shows a 25% discount. Another sign shows different white shirts with off. A third sign shows white shirts that cost 0.45 less than the original price.
Show Patrik which discount he should ask for in order to save the most money on a white shirt.
Show more than one way to solve this problem.

**13. **13 A Mini-Gallery Walk Find a partner group.
Share your group?s solutions with your partner group. Designate a reporter who will describe the different ways in which you solved the problem.
Listen as the other group?s reporter describes its solutions.
Compare the two groups? solutions. How are they similar? How are they different?

**14. **14 Session C ? Conceptual Development Warm Up ? A KWL Chart Know, Wonder, Learned
Quilting ? Problem #2
A Gallery Walk

**15. **15 Warm Up ? KWL ChartKnow, Wonder, Learned
Give each participant a copy of copy of BLM2Give each participant a copy of copy of BLM2

**16. **16 Quilting ? Problem #2 Ahmed and Tamara are sewing a quilt together. The finished quilt will be square and have 10 squares on each side. So far they have finished 0.56 of their quilt.
Their friends, Soumia and Carlos, are working on another quilt of the same size. They have finished of their quilt.
The friends want to know who has finished more.
Show the solution of this problem by using:
a 10 x 10 grid
two stacked number lines

**17. **17 A Gallery Walk

**18. **18 Session D ? Alternative Algorithms Warm-Up ? The Meaning of Ratio
Best Buy on Juice ? Problem #3
Bansho
Engaging in Rich Problems
Professional Learning Opportunities

**19. **19 Warm Up ? The Meaning of Ratio Take 3 stick-on notes.
Ask 3 people (from different tables) to share what ?ratio? means to them. Use words, symbols, pictures, or numbers.
Write or draw what you hear about ?ratio.?
Return to your table.
With your table group, look at all of the comments.
Collectively, write a definition and or representation of ?ratio.?

**20. **20 Best Buy on Juice ? Problem #3 Sandro and Julia need to buy boxes of juice for their camping trip. At one store, the cost is $27.60 for 24 boxes. At another store, 18 boxes cost $19.80. Their mother told them not to spend more than $1.12 per box.
a) Which is the better buy?
b) Where should they buy the juice?

**21. **21 Reflecting/Connecting Bansho: sorting and classifying the details in the solutions presented by participants
Bansho helps students:
?see what they need to do and think about;
? see connections between parts of the lesson,
concepts, and solutions;
? organize their thinking; and
? discover new ideas. Teachers can also use samples of student work from Eworkshop at www.eworkshop.on.caTeachers can also use samples of student work from Eworkshop at www.eworkshop.on.ca

**22. **22 Engaging in Rich Problems Rich problems:
can be represented with a variety of mathematics;
are grounded in a context meaningful to students;
inherently contain the mathematics that the teacher wants the students to learn;
have several entry points and are conducive to extensions, allowing for differentiated instruction; and
require students to use high-level thinking skills.

**23. **23 Professional Learning Opportunities Collaborate with other teachers through:
Co-teaching
Coaching
Teacher inquiry/study groups
View:
Coaching Videos on Demand (www.curriculum.org)
Deborah Ball webcast (www.curriculum.org)
E-workshop (www.eworkshop.on.ca) <in facilitator?s guide go to eWorkshop Success Factors>
<in facilitator?s guide go to eWorkshop Success Factors>